Download Current food web models cannot explain the overall topological

OIKOS 115: 97 109, 2006
Current food web models cannot explain the overall topological
structure of observed food webs
Jeremy W. Fox
Fox, J. W. 2006. Current food web models cannot explain the overall topological
structure of observed food webs. Oikos 115: 97 109.
Topological food webs illustrating ‘‘who eats whom’’ in different systems exhibit
similar, non-random, structures suggesting that general rules govern food
web structure. Current food web models correctly predict many measures of food
web topology from knowledge of species richness and connectance (fraction of possible predator prey links that actually occur), together with assumptions about the
ecological rules governing ‘‘who eats whom’’. However, current measures are relatively
insensitive to small changes in topology. Improvement of, and discrimination among,
current models requires development of new measures of food web structure. Here I
examine whether current food web models (cascade, niche, and nested hierarchy
models, plus a random null model) can predict a new measure of food web structure,
structural stability. Structural stability complements other measures of food web
topology because it is sensitive to changes in topology that other measures often miss.
The cascade and null models respectively over- and underpredict structural stability for
a set of 17 high-quality food webs. While the niche and nested hierarchy models provide
unbiased predictions on average, their 95% confidence intervals frequently fail to
include the observed data. Observed structural stabilities for all models are
overdispersed compared to model predictions, and predicted and observed structural
stabilities are uncorrelated, indicating that important sources of variation in structural
stability are not captured by the models. Crucially, poor model performance arises
because observed variation in structural stability is unrelated to variation in species
richness and connectance. In contrast, almost all other measures of food web topology
vary with species richness and connectance in natural webs. No model that takes
species richness and connectance as the only input parameters can reproduce observed
variation in structural stability. Further progress in predicting and explaining food
web topology will require fundamentally new models based on different input
parameters.
J. W. Fox, Dept. of Biological Sciences, Univ. of Calgary, 2500 University Dr. NW,
Calgary, AB, Canada, T2N 1N4 ([email protected]).
A food web is a synthetic description of an ecological
community. Even topological (qualitative) food webs,
which simply illustrate ‘‘who eats whom’’, summarize a
great deal of information on diversity, species composition, trophic structure, energy and material flows, and
species interactions (Williams and Martinez 2000). For
this reason ecologists have long sought patterns in food
web topology (Cohen 1978, Cohen and Newman 1985).
Patterns in food web topology suggest the operation of
general ‘‘rules’’ that generate these patterns. Discovery of
these rules would provide fundamental insight into the
ultimate causes of community structure. For instance, if
community structure ultimately reflects dynamic stability constraints, observed food web structures should be
more stable than expected by chance (Yodzis 1981).
There may also be surprising fundamental analogies
Accepted 21 March 2006
Subject Editor: Lennart Persson
Copyright # OIKOS 2006
ISSN 0030-1299
OIKOS 115:1 (2006)
97
between the rules governing food web structure and the
rules structuring other kinds of networks, like the
internet (Dunne et al. 2002). Knowledge of the rules
governing food web topology also would aid progress on
numerous applied issues, such as predicting the direct
and indirect impacts of invasive species (Woodward and
Hildrew 2001). Research on these issues will be challenging, because substantial progress likely will require
detailed empirical data, and models capable of reproducing observed details (Williams and Martinez 2000,
Borer et al. 2002).
Several models suggest detailed patterns in food web
structure may reflect physical and/or phylogenetic constraints on ‘‘who can eat whom’’ (Cohen and Newman
1985, Williams and Martinez 2000, Cattin et al. 2004).
For instance, in aquatic systems predators typically are
larger than but not too much larger than-their prey
(Warren and Lawton 1987, Cohen et al. 1993, 2003), and
in many systems phylogenetically-related species have
similar diets (Cattin et al. 2004). Current models
correctly predict many measures of food web topology
using only two input parameters, species richness (S) and
connectance (C), combined with ecological assumptions
intended to capture the effects of physical/phylogenetic
constraints on diet (Williams and Martinez 2000, Cattin
et al. 2004). Connectance is the fraction of possible
predator prey links that are actually observed, and so
equals 1 when there are S2 links (i.e. every species eats
every other species, including itself). The input parameters S and C together determine the number of links
in the food web, while the ecological assumptions govern
link arrangement. The use of S and C as input
parameters is motivated in part by the observation that
many patterns in food web structure take the form of
simple functional relationships between measures of
food web topology (dependent variables), and S and/or
C (independent variables) (Cohen and Newman 1985,
Williams and Martinez 2000, Cattin et al. 2004). Even
those measures of food web structure that are not wellpredicted by current models (e.g. numbers of ‘‘network
motifs’’, Bascompte and Melián 2005) vary systematically with species richness and connectance, suggesting that slightly modified versions of current models
might predict these measures as well. Indeed, the history
of theoretical work on food web topology is largely a
history of improving the ecological assumptions describing physical/phylogenetic constraints on link arrangement, so as to better predict various measures of food
web topology (Cohen and Newman 1985, Williams and
Martinez 2000, Cattin et al. 2004).
Continued progress in understanding food web topology depends on having measures of topology that, alone
or in combination, are sufficiently sensitive to provide
stringent tests of a model’s predictions. However, many
current measures are relatively insensitive to small
changes in food web structure. Current measures of
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food web topology lack sensitivity because they take
the form of averages, sums, standard deviations, or
proportions, calculated across all species or across all
subparts of a web. For instance, mean food chain length
is simply the average length of all linear food chains from
a basal species to a top predator, and comprises an
average across hundreds, thousands, or tens of thousands of food chains, depending on the complexity of the
web (Williams and Martinez 2000). Randomly shuffling
a few, or even many, of the links in a complex food web
would be unlikely to greatly alter mean food chain
length. A model could therefore correctly predict mean
food chain length without actually producing a food web
that closely resembles the real web.
One way around this problem is to test a model on
different, complementary measures of food web structure. However, current measures of food web structure
apparently are not sufficiently complementary to discriminate among similar models. The niche and nested
hierarchy models produce food webs that look quite
different to the naked eye (Fig. 1 in Cattin et al. 2004),
but make very similar predictions for most current
measures of food web topology (Cattin et al. 2004,
Stouffer et al. 2005). Current food web models dramatically outperform a random null model, and so clearly
capture at least some of the broad determinants of food
web topology, but which models (if any) also capture the
details is an open question that is difficult to answer with
current measures of food web topology.
(a)
(b)
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structural stability=–0.226
structural stability=–0.099
Fig. 1. Two hypothetical food webs of 9 species, differing only
in the placement of a single predator prey link (bold arrow).
Arrows point from prey to predator; looped arrows indicate
cannibalism. Webs (a) and (b) are identical in many topological
properties (including species richness; connectance; mean food
chain length; proportions of top, intermediate, and basal
species; proportion of omnivores; proportion of cannibals;
mean and standard deviation of generality and vulnerability;
degree distribution; see Williams and Martinez 2000 and Dunne
et al. 2002 for definitions) but differ greatly in structural
stability.
OIKOS 115:1 (2006)
Here I propose a new measure of food web topology,
structural stability, and ask whether current food web
models correctly predict structural stability. I first
express a topological food web in matrix form, and
then define structural stability as the largest real part of
the eigenvalues of the food web matrix. Structural
stability can be interpreted as the contribution of food
web topology (which determines the signs of species
interaction strengths) to local asymptotic (dynamic)
stability. In contrast, many proposed measures of food
web structure lack obvious ecological interpretations
(e.g. the standard deviation of the lengths of all linear
food chains in a web, Williams and Martinez 2000).
This ecological interpretability makes structural stability more versatile than a simple abstract measure of
food web structure. For instance, structural stability
links topological food web theory to the study of
dynamic constraints on food web structure (Yodzis
1981). Because structural stability is not a simple sum,
average, standard deviation, or proportion, it is sensitive to structural changes (repositionings of predator prey links) that do not alter other measures of food web
topology such as mean food chain length (Fig. 1).
Structural stability is not a perfect measure of food web
structure (no single number can summarize all aspects
of food web topology), but rather complements previously-proposed measures by capturing many structural changes that previous measures likely would miss.
I compared the structural stabilities of 17 observed
food webs to the predictions of a random null model,
and three food web models: the cascade model (Cohen
and Newman 1985), the niche model (Williams and
Martinez 2000), and the nested hierarchy model (Cattin
et al. 2004). These three food web models differ in the
details of their assumptions, but all share fundamental
similarities. All three assume that the physical/phylogenetic similarity of species can be summarized with a
single number (a ‘‘niche value’’), and that physical/
phylogenetic constraints tend to cause species with larger
niche values to consume those with smaller niche values
(a ‘‘trophic hierarchy’’). Further, the niche and nested
hierarchy models (but not the cascade model) assume
that species tend not to consume those with much
smaller niche values (Stouffer et al. 2005). However, all
three models differ in their details (Williams and
Martinez 2000, Cattin et al. 2004). Structural stability
provides an opportunity to test whether a model gets
both the basics and the details right, since unlike many
other measures of food web structure, structural stability
is not analytically derivable from the shared basic
assumptions of the niche and nested hierarchy models
(Stouffer et al. 2005). Structural stability therefore tests
the performance of current food web models in a new
way, by measuring aspects of food web topology not
considered previously.
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Methods
Construction of qualitative webs
I considered 17 highly-detailed qualitative natural food
webs: the 14 previously-published webs considered by
Dunne et al. (2002), plus three additional marine webs
(the Benguela system off South Africa, a small Caribbean coral reef, and a North American continental
shelf) described in Dunne et al. (2004). Dunne et al.
(2002) also considered two unpublished webs that I do
not consider. The 17 webs come from a variety of
terrestrial, freshwater, and marine communities, and
previous studies describe various features of these webs
(Williams and Martinez 2000, Dunne et al. 2002, 2004,
Cattin et al. 2004). The webs range in richness from 25 155 trophic species (species sharing the same predators
and prey). I considered trophic species so as to conform
to published food web models (Williams and Martinez
2000, Cattin et al. 2004). Aggregation of biological
species into trophic species reduces species richness to
an average of 85% of its original value, with 12/17 webs
having trophic species richness ]/90% of biological
species richness.
A qualitative food web can be expressed as a matrix A
where non-zero elements aij and aji (i,j /1,. . .,S, where S
is trophic species richness) indicate the existence of a
trophic interaction between species i and j. I assume that
aij / /1 if species j is a predator of i, and assign the
corresponding element aji a value of /1. Describing the
interaction between predator j and prey i with a pair of
matrix elements aij and aji of differing signs, rather than
with a single negative element aij, allows ecologicallymeaningful calculation of structural stability. If species i
and j consume one another (mutual predation), I assume
aij /aji / /1, thereby treating mutual predation as a
form of interference competition. I assign cannibalistic
species aii / /1, thereby treating cannibalism as intraspecific interference competition. Because I focus on
trophic structure, I assume no direct interactions among
species that are neither prey nor predator of one another
(aij /aji /0).
Measurement of structural stability and statistical
analyses
I define structural stability as the largest real part of the
eigenvalues of A (max(Re(l(A)))). In dynamical food
web theory, the matrix A is interpreted as giving the
interaction strengths of one species on another, and the
largest real part of the eigenvalues of A measures
asymptotic dynamic stability (resilience), the ability
of the system to recover from a small perturbation
(May 1973). In the context of dynamical food web
theory, when the non-zero elements of A all have the
same absolute magnitude (jaijj /1), the elements aij are
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interpreted as the signs of the interaction strengths (May
1973). May (1973) pointed out that systems with
appropriate sign structures will always recover from a
small perturbation, whatever the absolute magnitudes of
the elements of A; such systems are called sign stable.
Sign stability is an all-or-nothing property a system is
either sign stable, or not. However, when the elements of
A all have the same absolute magnitude, I suggest that
max(Re(l(A))) can be interpreted as a measure of the
contribution of qualitative (sign) structure to dynamic
stability (for related suggestions, see Dambacher et al.
2003a, 2003b). I call this contribution structural stability. Webs with high structural stability (i.e. low
max(Re(l(A))), when all non-zero aij have the same
absolute magnitude) will be dynamically stable for many
different choices of the quantitative magnitudes of the
aij. Conversely, webs with low structural stability would
require aij of very precisely chosen quantitative magnitudes in order to be dynamically stable (Dambacher
et al. 2003a, 2003b).
Structural stability also is a useful measure of food
web topology that is sensitive even to small changes in
topology. This is because structural stability reflects the
numbers of negative and positive feedback loops of
different lengths in a food web, as well as self-feedback
(non-zero diagonal elements), and feedback loops reflect
the food web topology (Neutel et al. 2002, Dambacher et
al. 2003b). Here, a feedback loop refers to a closed chain
of trophic effects (of predator on prey and/or prey on
predator) that does not include any species more than
once. Feedback loops include ‘‘loops’’ sensu Williams
and Martinez (2000) closed chains of effects of
predators on prey as well as other kinds of loops.
Negative feedback loops are those where the product of
the signs of the effects in the loop is /1 (e.g. a twospecies loop from predator i to prey j and back again has
ajiaij /(/1)(/1)/ /1). Positive feedback loops are
those where the product of the signs of the effects in
the loop is /1. Negative feedback loops increase
structural stability (i.e. reduce max(Re(l(A)))) and
positive feedback loops decrease stability, although
structural stability also depends on the frequency of
negative feedback loops of different lengths (Neutel et al.
2002, Dambacher et al. 2003b). Changing the position of
even one predator prey link in a food web will very
likely change the frequency of negative and positive
feedback loops of different lengths, thereby altering
structural stability (Fig. 1). Structural stability of course
does not capture all differences in food web topology,
since it is possible for two webs of different topologies to
have the same structural stability. However, I show below
that this possibility is unlikely; the more two webs differ
in topology, the more they are likely to differ in
structural stability.
Structural stability necessarily equals zero when all or
most diagonal elements aii /0. In order to study the
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qualitative structure of observed and model food webs
with B/ /10 cannibalistic species (a category which
includes 12/17 observed webs), it is therefore necessary
to assume that some of the non-cannibalistic diagonal
elements aii are non-zero. I assume that a randomlyselected 50% of the non-cannibalistic species in each
observed web have aii / /1, which can be thought of as
reflecting intraspecific competition. The remainder of
the non-cannibalistic species have aii /0. Because the
value of structural stability for any web will vary slightly
depending on which species have aii / /1, I evaluated
the structural stabilities of 100 realizations of each
observed web, each with a randomly-selected 50% of
non-cannibalistic diagonal elements equal to /1. For
each of the 17 webs, the mean of these 100 realizations
gives the ‘‘true’’ observed structural stability, which a
successful food web model will accurately predict. The
fraction 50% necessarily is an arbitrary choice. A limited
number of additional simulations indicated that results
were qualitatively unchanged when 10% or 90% of noncannibalistic species had aii / /1. Alternative treatments of the diagonal elements have serious drawbacks.
The alternative, biologically-motivated choice of setting
aii / /1 for all basal species (those with no prey) and
aii /0 for non-basal non-cannibals (Pimm and Lawton
1977) would make structural stability very sensitive to
the proportion of basal species in the food web. This
would therefore reward a model that correctly predicted
the proportion of basal species while poorly predicting
overall food web topology. The alternative arbitrary
choice of setting all aii / /1 would be undesirable for
two reasons. First, it would fail to reward models for
correctly predicting the number of cannibalistic species,
since aii would equal /1 whether or not species i was
cannibalistic. Second, structural stability necessarily
equals /1 for many low-connectance webs when all
aii / /1, independent of the topology of the offdiagonal elements. For instance, seven of the 10 lowestconnectance observed webs have structural stability of
/1 when all aii are set to /1. Setting all aii / /1 would
therefore prevent analysis of whether food web models
correctly reproduce the observed structural stabilities of
low-connectance webs. Randomly setting an intermediate fraction of non-cannibalistic diagonal elements to
/1 allows the topology of off-diagonal and cannibalistic
links to dictate structural stability.
I analyzed three food web models (cascade, niche and
nested hierarchy models), plus a random null model.
Cohen and Newman (1985), Williams and Martinez
(2000) and Cattin et al. (2004) describe the assumptions
of the food web models. I generated null model realizations by randomly shuffling the locations of the observed
effects of predators on prey, so that every null model
realization had exactly the same connectance C as the
corresponding observed web (note that C is always
calculated without including the negative diagonal
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elements randomly assigned to non-cannibalistic species). Since realized connectance can differ from that of
the corresponding observed web in the other models, for
these models I only used realizations having C within
9/3% of the corresponding observed web, as in Williams
and Martinez (2000). While it is possible to generate
cascade model webs having exactly the same C as the
corresponding observed webs, this is computationally
slower and would not have altered the results. Each
model provides the location of predator prey links
(negative effects of predator j on prey i, aij / /1). For
all models, every off-diagonal element aij / /1 was
assigned a corresponding aji //1, except when aij /
aji / /1, as in the observed webs.
All the models examined here are stochastic: for
given values of species richness and connectance, each
model can produce many different food web topologies.
Each observed food web can be thought of as one
realization of an unknown stochastic model, which is
then assigned 100 random sets of diagonal elements for
purposes of estimating its structural stability. Accordingly, comparing model predictions to the observations
requires generating many realizations of the model with
the same S and C as the corresponding observed web,
with each model realization then assigned 100 random
sets of diagonal elements so that its structural stability
can be determined. I generated 200 realizations of each
model corresponding to each of the 17 observed webs
(/17 /200/3400 realizations/model /4 models/
13 600 realizations). For each of the 13 600 model
realizations, I calculated structural stability by assigning 100 random sets of diagonal elements, in exactly the
same way as for the observed webs. The mean
structural stability of each realization, averaged across
the 100 diagonal element sets, is taken as the structural
stability of that realization, just as for the observed
webs. The overall mean of each set of 200 structural
stabilities defines a model prediction, to which the
corresponding observed structural stability is compared. The 2.5% and 97.5% points of the distribution
of 200 structural stabilities define a 95% confidence
interval around the model prediction, within which the
structural stability of the corresponding observed web
should fall 95% of the time (i.e. in /16 out of 17 cases)
if the model is successful (percentile confidence interval, Manly 1997). While more model realizations, and
more sets of diagonal elements/realization, would have
been desirable, calculation of eigenvalues for large food
webs is computationally expensive and time-consuming.
The ‘‘sampling effort’’ used proved adequate for
addressing the questions asked.
To facilitate comparison with previous work (Williams
and Martinez 2000), I also calculated normalized errors
for the model predictions. Normalized error is the raw
error (difference between the model prediction [overall
mean of 200 realizations] and the observed value),
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divided by the standard deviation of the 200 realizations.
An accurate model will have an overall mean normalized
error (averaged across all 17 webs) near zero. If
structural stabilities of a set of 200 realizations are
approximately normally distributed (an assumption not
made by the percentile confidence intervals), the standard deviation of these realizations defines a parametric
confidence interval for the model prediction. Ninety-five
percent of the 200 realizations should fall within 9/1.96
standard deviations of the mean, so that an observed
value within this range represents a statistically acceptable fit between prediction and observation. However,
normalized errors reward models for making highly
variable, imprecise predictions. To test this possibility, I
calculated the standard deviation of the normalized
errors for each model. The expected value of this
standard deviation is 1. Larger values indicate overdispersion the observed data are more widely dispersed around the model predictions than expected,
given the variability among the model realizations.
Overdispersion indicates that the model does not capture
all of the sources of variability in the observed data.
I checked all model realizations for empty species
(species with no predators or prey), and for identical
species (species with the same predators and prey),
replacing such realizations with new realizations. I
lacked software to check whether any realization actually comprised two or more disconnected subwebs without containing empty species. Previous studies of other
aspects of qualitative food web structure do not test for
multi-species disconnected subwebs because such webs
likely are very rare (R. Williams, pers. comm.). Realizations comprising disconnected multi-species subwebs are
most likely to occur at low connectance, but low
connectance webs also are the most likely to contain
empty species and so be eliminated from the analysis.
Spot checks failed to reveal any realizations comprising
two or more disconnected subwebs. Note that one of the
observed webs (Grassland) actually comprises disconnected multi-species subwebs (R. Williams, pers.
comm.). I followed previous studies (Williams and
Martinez 2000, Dunne et al. 2002) and included the
Grassland web in the analyses. Dropping this web from
the analyses does not qualitatively alter any of the
results, and produces only minor quantitative changes
(not shown).
To further explore the relationship between observed
and predicted values of structural stability, I calculated
the parametric correlation coefficient r between the
observed structural stabilities, and the predicted values
from each of the four models. This analysis asks whether
variation in model predictions mirrors variation in the
observed data, even if the model predictions are
inaccurate and/or the observed data are overdispersed.
A positive correlation suggests that the model captures
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at least some of the sources of variation in the observed
data.
To better understand the source of any inaccuracy in
model predictions, I tested whether both observed and
predicted structural stabilities vary with S and C, the
input parameters in the food web models. If most of the
variation in observed structural stability is unrelated to
S and C, a model that takes S and C as the only input
parameters will have difficulty accurately and precisely
reproducing the observed data. However, systematic
variation in observed structural stability with S and C
that is not accurately reproduced by current models
might be reproduced by a modified model. To test
whether structural stability depends on S and C in
different ways in observed vs model webs, I first
performed an analysis of covariance (ANCOVA) for
the effects of web type (observed, cascade, niche, nested
hierarchy, or null), S, C, S2, C2, and SC (continuous
covariates), and the interactions between web type and
the covariates on structural stability (n /85 /5 web
types /17 webs). I performed this analysis using
predicted structural stabilities (overall means of 200
realizations/web), rather than using all realizations, so
as to equalize sample sizes between observed and
model data. However, inclusion of all model realizations in the ANCOVA did not qualitatively affect the
results (not shown). I chose S, C, S2, C2, and SC as the
covariates based on preliminary inspection of the data.
The effects of several interaction terms were highly
significant, indicating that the effects of the covariates
were web type-dependent. To interpret this type-dependency of the effects of the covariates, I conducted
separate multiple regressions of structural stability on
S, C, S2, C2, and SC for each web type, and used a
stepwise procedure to select the minimal adequate
model.
To aid ecological interpretation of the results, I
counted the numbers of cannibalistic links and mutual
predation loops in each observed web and model
realization. Variation in the frequency of cannibalism
(a form of negative feedback) and mutual predation (a
simple positive feedback loop) might partially explain
variation in structural stability.
I constructed all food web model realizations using
Microsoft Excel 2002 Visual Basic macros provided by
L.-F. Bersier. I added diagonal elements for noncannibalistic species, calculated eigenvalues, and conducted statistical analyses using R for Windows 2.1.0 (R
Development Core Team 2004). Construction of all
1 361 700 food web matrices (4 models/17 food
webs /200 realizations/model/web/100 random diagonal element sets/realization, plus 17 observed webs /100
random diagonal element sets/web) and calculation of
their eigenvalues required several hundred hours of
computing time.
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Testing whether structural stability sensitively
measures overall food web topology
Illustrative examples like Fig. 1 suggest that structural
stability generally is sensitive to small changes in food
web topology, but do not provide formal evidence.
Further, such examples also suggest that structural
stability might be overly sensitive structural stability
might be such a complex function of food web topology
that two food webs differing greatly in overall topology
might be as likely to have similar structural stabilities as
two topologically-similar food webs. Next I describe a
test for whether the expected difference in structural
stability between food webs increases monotonically as
the true difference in topology between the two webs
increases.
Conducting this test requires a formal measure of the
‘‘true’’ difference in topology between two food webs.
However, defining a measure of the ‘‘true’’ difference in
topology between two food webs is not straightforward.
One plausible measure of the true difference in topology
between two food webs with identical S and C is the
minimum number of predator prey links that would
have to be repositioned in order to make the two webs
topologically identical. I tested whether the expected
difference in structural stability between webs is monotonically related to this measure of the true difference in
food web topology. For each of the nine observed
food webs in my dataset with SB/62, I generated
partially-randomized versions with k/1,2,. . .,20 randomly-selected links randomly repositioned (100 partially-randomized versions of each web for each value of
k). Generating these data for observed webs with ]/62
species would have been extremely time-consuming, but
there is no reason to think that results for these nine
observed webs are unrepresentative. To randomly-reposition a link, I deleted the original randomly-selected
link, and replaced it by creating a new link between two
randomly-chosen species that were unlinked in the
original observed web. I then calculated the structural
stability of each partially-randomized version of each
web as for the observed webs. For each observed web, the
mean structural stability of the 100 partially-randomized
versions with k links repositioned gives the expected
structural stability of a web k links removed from the
observed web. These data test whether structural stability
is sensitive to small changes in food web structure, since
/ /20
links. To obtain
all observed webs comprise additional data on webs with many links repositioned,
I used the predicted structural stabilities of webs
generated by the null model. In the null model, a
proportion 1-C of the randomly-placed links are expected to link predators and prey which are unlinked in
the corresponding observed web. That is, webs generated
by the null model have the same structure as webs in
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which a fraction 1-C of the links are randomly repositioned.
Results indicate that structural stability is a good
index of food web topology. For eight of nine webs,
expected structural stability increases significantly with
an increasing proportion of links repositioned, reaching
an asymptote once a very high proportion of links are
repositioned (Fig. 2). One of the webs (Grassland) has a
near-random topology, so that random repositioning of
links does not substantially alter its expected structural
stability (Fig. 2i). The regressions remain significant for
seven of eight webs if data from the null model webs are
dropped from the analysis, indicating that expected
structural stability is sensitive even to relatively small
changes in the true food web topology. These results
show that two webs with the same S and C and similar
topologies are likely to have similar structural stabilities,
while two webs with the same S and C but very different
topologies are unlikely to have similar structural stabilities. Model predictions (each of which is the mean of
200 model realizations) should be accurate if the model
tends to generate food webs that are topologicallysimilar to the corresponding observed webs.
However, the results shown in Fig. 2 do not entirely
rule out the possibility that a food web model also could
attain high accuracy while generating webs with very
different topologies than those observed. This is because
food web models do not generate a random sample of all
the food web topologies that could be generated for
given values of S and C. For instance, imagine it were the
case that structural stability increases with the proportion of cannibalistic species in a web, but decreases with
the proportion of omnivores (this example is purely
hypothetical, and merely intended as a simple illustration of the point at hand). In this case, a food web model
that generated webs with more cannibals and fewer
omnivores than observed might accurately reproduce
observed structural stabilities while incorrectly predicting many other features of food web topology. For this
reason, structural stability provides a conservative test
for structural differences between observed and model
food webs; it is possible that a food web model might
predict ‘‘the right structural stability for the wrong
reasons’’. Testing food web models with other measures
of food web structure provides a partial check on this
possibility (Williams and Martinez 2000, Cattin et al.
2004): a model that correctly predicts many different,
complementary measures of food web topology likely
generates food webs that closely resemble observed webs.
However, current models do not consistently reproduce
observed structural stabilities, so the theoretical possibility that they might consistently predict ‘‘the right
structural stability for the wrong reasons’’ is moot.
Note that my proposed measure of the ‘‘true’’
difference in food web topology cannot be used to
directly compare observed food webs and webs generated
1.2
1.2
0.4
a
b
c
0.6
0
0
–0.3
OIKOS 115:1 (2006)
Mean structural stability
Fig. 2. Mean structural
stabilities of partiallyrandomized versions of nine
food webs (a) Bridge Brook; (b)
Coachella Valley; (c) Skipwith
Pond; (d) Chesapeake Bay; (e)
Benguela; (f) St. Martin Island;
(g) St. Marks Seagrass; (h)
Small Reef; (i) Grassland, vs
the proportion of links
randomly-repositioned. Error
bars are9/1 SD. Fitted curves
ax
are of the form y
c
bx
and when shown are
statistically significant at the
a/0.05 level.
–0.4
0
0
0.1
0.45
0.9
0
0.8
d
0.4
0
0.8
0.3
e
0.4
0.8
0.5
1
0.5
1
f
0
0
0
–0.3
–0.3
–0.2
0
0.2
0.5
0
1
1
g
0.4
0.8
0
0.2
h
i
0
0
0
–0.2
–0.4
0
0.5
1
–0.2
0
0.45
0.9
0
Proportion links repositioned
103
by the cascade, niche, and nested hierarchy models, since
these models assume that species are ranked along a
‘‘niche axis’’. In practice, the true ranking of species in
an observed food web is unknown, and so the number of
link repositionings separating an observed web from a
model-generated web also is unknown. Therefore, indices of food web topology are required to test food web
models (Solow 2005). To my knowledge, structural
stability is the first such index to be rigorously validated
by comparison with a formal measure of the true
difference in topology (Fig. 2).
0.3
(a)
1
1:
0
–0.4
0.8
(b)
Results
0.4
Predicted structural stability
The 17 observed webs vary significantly in structural
stability, much more than expected given the random
variation in structural stability among the 100 different
sets of random diagonal elements assigned to each web
(ANOVA, F16,1683 /178.40, PBB
/ /0.001). This indicates
that there is substantial variation in observed structural
stability, which a successful food web model should
reproduce. Variation in observed structural stabilities is
not simply noise resulting from random assignment of
diagonal elements.
The four models examined here vary in accuracy.
Overall, the cascade model tends to underestimate
structural stability (i.e. predicts max(Re(l(A))) less
than that observed), while the null model tends to
overestimate structural stability (Table 1). These results
hold whether overall mean accuracy is measured using
normalized or raw errors (Table 1). The niche and nested
hierarchy models are broadly accurate on average, at
least when accuracy is measured in terms of raw error
(Table 1).
However, none of the four models is as accurate as
would be expected for a model identical with the
unknown model that generated the observed data.
Ninety-five percent percentile confidence intervals for
structural stability predicted by the cascade model
include the observed data for only 5 of 17 webs, rather
than the expected 16 of 17 (Fig. 3a). Corresponding
results for the other models are: niche model, 13 of 17
webs (Fig. 3b); nested hierarchy model, 8 of 17 webs
(Fig. 3c); null model, 10 of 17 webs (Fig. 3d). Similar
results hold when parametric confidence intervals are
1:1
0
–0.4
1.2
(c)
0.8
1:1
0.4
0
–0.4
1.6
(d)
1.2
0.8
1:1
0.4
0
–0.4
–0.4
Table 1. Mean raw error (RE), mean normalized error (NE),
and standard deviation of NE for each model; n/17 for each
model.
Model
Mean
RE
Mean
NE
Standard
deviation of NE
Random
Cascade
Niche
Nested hierarchy
0.26
/0.17
0.05
/0.05
1.02
/10.54
/0.02
/1.49
1.91
12.31
1.58
2.25
104
0
Observed structural stability
0.4
Fig. 3. Observed vs predicted structural stabilities for the
cascade model (a), niche model (b), nested hierarchy model
(c), and null model (d). Error bars are 95% percentile confidence
intervals. Bold dashed lines indicate 1:1 relationships. Failure of
an error bar to overlap the dashed line indicates a significant
difference between predicted and observed values at the a /0.05
level. Filled symbols highlight two food webs (Coachella Valley
and Skipwith Pond) with very similar species richness and
connectance but very different observed structural stabilities.
OIKOS 115:1 (2006)
OIKOS 115:1 (2006)
0.3
(a)
0
–0.4
0.4
(b)
0
Structural stability
used. For the cascade, niche, nested hierarchy, and null
models, parametric 95% confidence intervals include 5 of
17 webs, 12 of 17 webs, 9 of 17 webs, and 12 of 17 webs,
respectively.
The four food web models also vary in their precision.
Ninety-five percent percentile confidence intervals are
considerably narrower on average for the cascade model
than the other models (Fig. 3), and the same is true for
parametric confidence intervals (not shown). For given
values of the input parameters S and C, the niche, nested
hierarchy, and null models often generate 95% confidence intervals that include structural stabilities far
outside the range of those observed in any natural
food web (Fig. 3). This contrasts with the relatively
greater precision of these models when predicting other
measures of food web topology (compare Williams and
Martinez 2000), and illustrates the sensitivity of structural stability to small changes in food web topology.
Different realizations of the same stochastic food web
model often exhibit similar values for many measures of
food web topology, but very different values of structural
stability (Fig. 3, Williams and Martinez 2000, Cattin
et al. 2004).
Despite this lack of precision in model predictions,
observed structural stabilities are actually more variable
than expected from any food web model. Standard
deviations of normalized errors for all for food web
models are /1, indicating overdispersion of the observed
data compared to model predictions (Table 1). The
especially high overdispersion for the cascade model
reflects the fact that its predictions are both precise and
inaccurate.
Predicted and observed structural stabilities are uncorrelated for all four models (all jrj B/0.17, all P /0.2;
Fig. 3), in contrast to the positive correlations between
predicted and observed values for most previouslyproposed measures of food web topology (Williams
and Martinez 2000, Cattin et al. 2004). Observed and
predicted structural stabilities are uncorrelated because
all models predict that structural stability varies systematically with S and C, while the observed data lack such
variation (Fig. 4). Removing the effect of web type and
associated interactions from the ANCOVA leads to a
highly significant reduction in fit (difference in residual
df /24, difference in residual SS/3.37, F /31.70, PB/
0.001), indicating that structural stability varies with
S, C, SC, S2, and C2 in different ways for different
web types. In a multiple regression, observed structural
stability does not vary significantly with S, C, SC,
S2, and C2 (R2 /0.32, P/0.45). In contrast, stepwise
multiple regression finds that predicted structural
stability varies significantly with some combination of
C, S, C2, and SC in all models (all R2 /0.90, all PB/
0.001). Inspection of residuals indicated conformity
with statistical assumptions for the ANCOVA and
all multiple regressions. While P values of multiple
–0.3
0.3
(c)
0
–0.3
0.8
(d)
0
–0.4
0
0.4
Connectance
Fig. 4. Observed and predicted structural stabilities (filled and
open symbols, respectively) vs connectance for the cascade
model (a), niche model (b), nested hierarchy model (c), and null
model (d). Predicted values vary systematically with connectance (either linearly or non-linearly), while observed values do
not. Observed values are shown in all panels to ease comparison
with predicted values. Error bars for predicted values are
omitted for clarity (see Fig. 3 for error bars).
105
regression models selected by a stepwise procedure
should be interpreted cautiously (Philippi 1993), the
clear conclusion is that structural stability depends in
some fashion on S and C only in models, not in the
observed data.
Observed webs have an average of 9.7 mutual predation loops. Null model realizations exhibit an average
of /100 mutual predation loops, averaging across all
realizations of all webs, while observed, niche, and
nested hierarchy realizations average 6 10 mutual predations. Mutual predation is impossible in the cascade
model. Variation in the number of mutual predation
loops is broadly associated with variation in structural
stability. Null model realizations are least-structurally
stable on average (i.e. highest values of max(Re(l(A)))),
while cascade model realizations are most structurally
stable, as expected if mutual predation loops (a form of
positive feedback loop) generate instability. However,
variation in the frequency of mutual predation loops
(/number of mutual predation loops divided by the
number of possible mutual predation loops) does not
appear to explain variation in structural stability among
observed webs, except that the Coachella Valley web is
both highly-unstable and has by far the highest frequency of mutual predation loops.
On average, 15% of species in observed food webs are
cannibalistic. Frequency of cannibalism is unrelated or
weakly related to structural stability. Cascade model
realizations lack cannibalism (a negative feedback loop),
but are more structurally stable than observed webs or
realizations of other models (Fig. 3, Table 1). Null model
realizations have similar frequencies of cannibalism to
observed webs (Williams and Martinez 2000), but are
much more unstable (Fig. 3, Table 1). Among observed
webs, frequency of cannibals is uncorrelated with
structural stability (P /0.21).
Discussion
Structural stability in observed and model food webs
None of the food web models examined here accurately
and precisely predicts the structural stabilities of observed food webs, implying that none of these models
successfully reproduces overall food web topology. The
niche and nested hierarchy model are at least unbiased,
but both make very imprecise predictions. For both
models, the 95% confidence intervals frequently include
values of structural stability far outside the observed
range. This lack of precision reflects the sensitivity of
structural stability to small changes in food web
topology. No stochastic model will make extremely
precise predictions of a highly sensitive measure of
food web structure. For this reason, lack of precision
per se is not a concern, so long as a model’s 95%
confidence intervals include the observed data 95% of
106
the time, and so long as the observed data are not more
variable than expected from the model. Unfortunately,
the niche and nested hierarchy models do not satisfy
either of these criteria. The 95% confidence intervals for
the niche model include the observed structural stability
data only 70 76% of the time, depending on whether
percentile or parametric confidence intervals are used.
The nested hierarchy model performs even worse in this
respect. Observed structural stabilities also are overdispersed compared to the predictions of current food
web models.
Even some inaccuracy and overdispersion need not
indicate serious failings in a model. A model that
captures some, but not all, of the sources of variation
in the data would be expected to exhibit some inaccuracy
and overdispersion, but its predictions would be expected to correlate positively with the observed data.
However, predicted and observed structural stabilities
are uncorrelated for all four models. Inaccuracy, overdispersion, and lack of correlation between observations
and predictions together indicate that the underlying
sources of variation in the model predictions do not
match the underlying sources of variability in the
observed data.
Although the the best current food web models do not
perform as well as might be hoped, they do outperform a
random null model. Random null model predictions are
highly biased, while those of the best current food web
models (niche and nested hierarchy models) are not.
Further, the random null model’s predictions are less
precise on average (broader confidence intervals) than
those of the niche or nested hierarchy models. The
broader confidence intervals of the random null model
explain why its confidence interval coverage is as good as
that of the nested hierarchy model. The equal accuracy
of the random and nested hierarchy models, as measured
by confidence interval coverage, should not be taken to
indicate that the random null model reproduces observed food web topologies as well as the nested
hierarchy model.
It is already known that current food web models,
although much better than a random null model, are not
perfect (Williams and Martinez 2000, Cattin et al. 2004,
Bascompte and Melián 2005). However, it might be
thought that by refining the ecological assumptions of
current models, their predictions could be improved.
After all, the most successful current models (the niche
and nested hierarchy models) were developed by refining
the ecological assumptions of the earlier cascade model
(Williams and Martinez 2000, Cattin et al. 2004).
Crucially, my results indicate that such refinement is
not possible, because the cause of poor model performance is not the ecological assumptions about the
nature of physical/phylogenetic constraints. Current
models predict structural stability poorly because current models take species richness and connectance as the
OIKOS 115:1 (2006)
only input parameters. All current models therefore
predict (i) that two webs with similar species richness
and connectance will have similar structural stabilities,
and (ii) that structural stability varies systematically with
species richness and connectance. But two observed webs
with similar species richness and connectance can vary
greatly in structural stability, and observed structural
stability does not vary significantly with species richness
and connectance, implying that observed structural
stability cannot be predicted solely from knowledge of
species richness and connectance.
The Coachella Valley and Skipwith Pond webs provide
a particularly striking illustration of this point. These
two webs have very similar species richness (S /29 and
25, respectively) and connectance (C /0.312 and 0.315,
respectively). According to all four models, they should
have similar structural stabilities (Fig. 3). However,
Coachella Valley is the most structurally unstable web
in the dataset, while Skipwith Pond is the third-most
stable. The structural stabilities of observed food
webs do not arise solely, or even mainly, from variation
in species richness and connectance. In this respect,
structural stability complements many other measures of
food web structure, which do vary significantly with S
and C.
Solow (1996) and Williams and Martinez (2000) noted
that several other features of food web topology (e.g.
mean food chain length) are overdispersed compared to
the predictions of the random, niche, and/or cascade
models (see also Murtaugh and Kollath 1997, Solow
2005). However, previous work evaluates model overdispersion using low-quality food web data (Solow
1996), considers overdispersion outside the context of
testing food web models (Murtaugh and Kollath 1997),
or focuses on the relative overdispersion of different
models (Williams and Martinez 2000). Further, previous
studies of overdispersion have focused on measures of
food web structure that vary with species richness and
connectance in observed webs, suggesting that the overdispersion might be corrected by refining the models’
ecological assumptions. No previous work demonstrates
overdispersion of observed data relative to model predictions that is not correctable by refining ecological
assumptions.
Identifying covariates that explain some of the variation in structural stability not captured by current food
web models might suggest new directions for model
development. Unfortunately, identifying such covariates
may prove difficult. For instance, it has been suggested
that food web structure might vary systematically among
habitats (Murtaugh and Kollath 1997). However, classifying observed webs by habitat (freshwater, terrestrial,
or marine) does not explain a significant amount
of variation in observed structural stability (ANOVA,
F2,14 /1.01, P/0.39). Observed webs could of course be
classified into a greater number of more finely-detailed
OIKOS 115:1 (2006)
habitat categories, and could be cross-classified by other
characteristics (e.g. primary productivity). However, the
limited number of observed food webs available would
severely limit the power of any statistical test based on
such detailed classification.
In summary, none of the four food web models
examined here capture the main sources of variation in
observed structural stability. More importantly, their
shortcomings cannot be corrected without changing the
identity of the input parameters. My results strongly
suggest that further increasing our understanding of
food web topology will require new models that use
different and/or additional input parameters besides
species richness and connectance (Loeuille and Loreau
2005).
Mutual predation and cannibalism
Variation in the frequency of cannibalism does not
explain variation in structural stability. I assumed that
all cannibalistic species, and half of other species,
exhibited stabilizing self-damped growth, so models
producing cannibalistic species more frequently should
be more structurally stable, all else being equal. This is
not the case, presumably because different models
produce food webs that differ in many respects besides
cannibalism frequency. Lack of an association between
observed cannibalism frequency and structural stability
may reflect the fact that most observed food webs have
very few cannibals. Variation in the frequency of mutual
predation loops explains some variation in structural
stability, but leaves most variation unexplained. Most
variation in structural stability apparently reflects more
subtle aspects of food web topology. It would be
interesting to examine the relative frequency of positive
and negative feedback loops of various lengths in
observed vs model food webs, as this should strongly
affect structural stability (Neutel et al. 2002, Dambacher
et al. 2003b).
Are observed food webs dynamically stable?
Although testing whether observed food webs are more
dynamically stable than expected by chance was not the
main goal of this work, structural stability does isolate
one component of dynamic stability, the sign structure of
the interaction matrix A (May 1973, Dambacher et al.
2003a,b). Consistent with many previous studies (Yodzis
1981, de Ruiter et al. 1995, Neutel et al. 2002, Emmerson
and Raffaelli 2004), I found that observed webs are
much more stable than null model webs. This shows that
the conclusions of Yodzis (1981) were not artifacts of
examining poor-quality food web data. My results
complement previous work in that I isolated the
contribution of food web topology to stability. Previous
107
studies examine only the relationship between quantitative structure and dynamic stability, where quantitative
structure refers to the absolute magnitudes of the
elements of A, or do not separate the effects of
qualitative and quantitative structure (de Ruiter et al.
1995, Neutel et al. 2002, Emmerson and Raffaelli 2004).
It would be interesting to analyze quantitative food webs
using both the methods used here, and those of previous
studies, in order to determine the extent to which any
dynamic stability arises from quantitative vs qualitative
aspects of food web structure.
Webs generated by food web models also are more
stable than null model webs. Since all three food web
models assume some form of ‘‘trophic hierarchy’’, such a
hierarchy apparently is stabilizing. However, the assumption of the niche and nested hierarchy models that
predators tend not to consume prey of much lower rank
than themselves, and may consume prey of higher rank,
is apparently destabilizing. These two models produce
less stable webs than the cascade model, which lacks
these assumptions.
The ‘‘trophic hierarchy’’ in all three food web models
is thought to reflect physical/phylogenetic constraints on
‘‘who can eat whom’’ (Warren and Lawton 1987, Cohen
et al. 1993, 2003, Cattin et al. 2004). This suggests the
interesting hypothesis that dynamic stability may be a
coincidental by-product of physical/phylogenetic constraints, rather than itself acting as a constraint on food
web structure. However, since none of the three food web
models accurately predicts observed structural stabilities,
none can be said to accurately capture any stabilizing
effects of physical/phylogenetic constraints.
Caveats
As with all food web studies, my conclusions depend on
the quality of the data. Even recently-published webs,
generated for the purpose of testing food web theory,
use various lines of evidence (some less reliable than
others) to document feeding links, lump together or
ignore numerous biological species (especially microbes,
parasites, and pathogens), and treat species with complex life histories in an ad hoc fashion (e.g. treating
different life history stages of the same biological
species as separate ‘‘trophic’’ species). While it is hard
to imagine that better data would reveal observed food
webs to have random structures, better data might be
either more or less similar to published models than
current data.
The results might also change if different assumptions
were made about how to express a food web as a matrix
A (Cohen et al. 1990). However, as long as the same
assumptions are used to convert both observed and
model food webs into matrices, it seems likely that
108
current food web models would fail to predict observed
structural stabilities.
Conclusions and future directions
Current food web models are impressive achievements. It
is by no means obvious that any simple model could
reproduce many of the topological features of complex
food webs (Caswell 1988). However, many of the
measures of food web topology examined previously
are relatively ‘‘easy targets’’, in that they are relatively
insensitive to small changes in food web topology.
Precisely and accurately predicting more sensitive measures of food web topology is a more challenging
task, one at which current food web models have little
success.
Because observed variation in structural stability is
unrelated to variation in species richness and connectance, any model that takes species richness and
connectance as the only input parameters will not
reproduce observed variation in structural stability.
Efforts to improve food web models via small refinements while retaining the same fundamental assumptions regarding species richness and connectance have
therefore reached a point of diminishing returns. My
results raise the possibility that significant further
improvements may even be impossible. Perhaps
each natural food web is a special case except for the
fact that, broadly speaking, species can be ranked in a
‘‘trophic hierarchy’’ and tend to consume species of
smaller, but not too much smaller, rank (Stouffer et al.
2005).
I suggest that the way to discover if further improvement is possible is to develop fundamentally new models
that sacrifice some of the simplicity of current models for
increased realism. It would be particularly interesting to
develop a food web model that does not require
connectance to be specified as an input parameter.
Treating connectance as an input parameter is common
practice in other areas of network theory (Dunne et al.
2002), but seems artificial when the network in question
is a food web. In food webs, connectance is a dependent
rather than an independent variable which reflects the
identities of the species and the factors controlling their
diets (Fox and McGrady-Steed 2002). The ultimate goal
of future modeling should be to not only correctly
predict food web structure, but to make the right
predictions for the right biological reasons. Achieving
this goal likely will require sacrificing some of the
simplicity of current models, but with the benefit of
allowing a greater variety of powerful and informative
comparisons with empirical data (including experimental
data; Fox and McGrady-Steed 2002). The model of
Loeuille and Loreau (2005) provides an intriguing step
in this direction.
OIKOS 115:1 (2006)
Acknowledgements Thanks to Jennifer Dunne for providing
the food web data and to Louis Bersier for providing the Excel
macros to generate the model webs. The ms benefited greatly
from discussion with Ed McCauley and members of the
McCauley lab. Owen Petchey pointed out a statistical error in
an earlier version of the ms.
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